Move particles first and stick to texture target
The Texture force field isn’t how it sounds - it’s actually a means of having fine control over the vectors within the force field by using the RGB colors a bit like a normal map with 0.5,0.5.0.5 being no force and +/- that being strength in each of the XYZ directions - rather than being a force that acts towards a texture. 

Probably the best solution would be to create your logo out of geometry (rather than an image - or hidden geometry that sits behind the actual image) and give this a force over its entire surface to attract to the logo, keyframing the force strength to activate it at a specific frame.

This can achieve something like the following :



The 'Text' logo includes a Force forcefield (set to Surface) - initially keyframed to zero strength.

Once the particles have fallen and collected on the floor the gravity field weight is keyframed down to zero (to allow the particles to float) and the Force on the logo is increased to a large negative value to draw the particles. A Drag forcefield slows down the particles to prevent them getting out of hand.

As the particles get closer to the logo the Force can be further reduced to allow the particles to settle down.

Blend file included